Oscar is Junior's dog. He is "only an adopted stray mutt, but he was the only living thing that [Junior] could depend on" (9). Oscar is actually not a main character of the story. He is introduced, sadly, to illustrate a bigger point about poverty.
Junior explains that the worst thing about being poor is not actually hunger. He then goes on to tell the story of Oscar, his adopted and beloved dog, who was "better than any human [he] had ever known" (9). But then Oscar gets sick and there's not enough money to take him to the vet. Instead of treating the sick dog, Junior's dad gets out his gun and puts him down.
For Junior, the painful memory of Oscar and his fate is an illustration of the real physical pain of poverty. In a sense, Oscar is a piece of the bigger picture compelling Junior to leave the reservation and do something more with his life.